Question

In a regression analysis involving 30 observations, the following estimated regression equation was obtained.

ŷ = 17.6 + 3.8x1 − 2.3x2 + 7.6x3 + 2.7x4

For this estimated regression equation, SST = 1,815 and SSR = 1,780. (a) At α = 0.05, test the significance of the relationship among the variables.

State the null and alternative hypotheses.

H0: β0 = β1 = β2 = β3 = β4 = 0

Ha: One or more of the parameters is not equal to zero.

H0: One or more of the parameters is not equal to zero.

Ha: β0 = β1 = β2 = β3 = β4 = 0

H0: One or more of the parameters is not equal to zero.

Ha: β1 = β2 = β3 = β4 = 0

H0: β1 = β2 = β3 = β4 = 0

Ha: One or more of the parameters is not equal to zero.

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.) p-value =

State your conclusion.

Do not reject H0. We conclude that the overall relationship is not significant.

Do not reject H0. We conclude that the overall relationship is significant.

Reject H0. We conclude that the overall relationship is significant.

Reject H0. We conclude that the overall relationship is not significant.

Suppose variables x1 and x4 are dropped from the model and the following estimated regression equation is obtained.

ŷ = 11.1 − 3.6x2 + 8.1x3

For this model, SST = 1,815 and SSR = 1,725.

(b) Compute SSE(x1, x2, x3, x4). SSE(x1, x2, x3, x4) =

(c) Compute SSE(x2, x3). SSE(x2, x3) =

(d) Use an F test and a 0.05 level of significance to determine whether x1 and x4 contribute significantly to the model. State the null and alternative hypotheses.

H0: One or more of the parameters is not equal to zero.

Ha: β1 = β4 = 0 H0: β1 = β4 = 0 Ha: One or more of the parameters is not equal to zero.

H0: One or more of the parameters is not equal to zero.

Ha: β2 = β3 = 0

H0: β2 = β3 = 0

Ha: One or more of the parameters is not equal to zero.

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.) p-value =

State your conclusion.

Reject H0. We conclude that x1 and x4 contribute significantly to the model.

Do not reject H0. We conclude that x1 and x4 contribute significantly to the model.

Reject H0. We conclude that x1 and x4 do not contribute significantly to the model.

Do not reject H0. We conclude that x1 and x4 do not contribute significantly to the model.

Answer 1

a)

H0: β0 = β1 = β2 = β3 = β4 = 0

Ha: One or more of the parameters is not equal to zero

SSE =SST-SSR=		35.0
MSR=SSR/k=		445.0
MSE=SSE/(n-k-1)=		1.4
F =MSR/MSE =		317.86
p value=0.0000

Reject H0. We conclude that the overall relationship is significant.

b)

SSE(x1,x2,x,3,x4) =

35

c)

SSE(x2x3) =

90

d)

H0: β1 = β4 = 0 Ha: One or more of the parameters is not equal to zero.

F =	((90-35)/2)/(35/25)=		19.64
p value =0.000

Reject H0. We conclude that x1 and x4 contribute significantly to the model.